Pentagon Area Calculator – Regular and Irregular Pentagons
A pentagon is a five-sided polygon. When all five sides are equal and every interior angle measures 108°, the shape is called a regular pentagon. When the sides and angles differ, the shape is an irregular pentagon. This calculator handles both cases and supports multiple input methods, unit conversion, and step-by-step results.
Regular Pentagon Formulas
A regular pentagon can be fully described by a single measurement. Three common measurements each lead to a distinct but equivalent area formula:
From Side Length (a)
The most common input is the side length a. The exact closed-form formula is:
A = (a² / 4) × √(25 + 10√5) ≈ a² × 1.720477
For example, a regular pentagon with side 5 m has an area of approximately 43.01 m². The perimeter is simply P = 5 × a, and the diagonal connecting two non-adjacent vertices equals d = a × φ, where φ ≈ 1.61803 is the golden ratio.
From Circumradius (R)
The circumradius R is the radius of the circle that passes through all five vertices (circumscribed circle). Given R:
A = (5/2) × R² × sin(72°)
The side length can be recovered with a = 2R × sin(36°), and the inradius with r = R × cos(36°).
From Inradius (r)
The inradius r (also called the apothem) is the distance from the center to the midpoint of any side — the radius of the inscribed circle. Given r:
A = 5 × r² × tan(36°)
The inradius formula is particularly useful in architecture and land surveying, where the distance to a boundary midpoint is easier to measure than a vertex distance.
Irregular Pentagon — Shoelace Formula
When the pentagon is not regular, providing five (x, y) vertex coordinates allows the calculator to apply the Shoelace formula (also called the Gauss area formula):
A = (1/2) × |Σ (xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ)| for i = 1…5 (vertex 6 = vertex 1)
Enter the vertices in order — either clockwise or counter-clockwise — and ensure the pentagon does not self-intersect. The formula works for any simple (non-self-intersecting) polygon, regardless of convexity. If the computed area is zero, the vertices are collinear or a duplicate exists.
Secondary Properties of a Regular Pentagon
Beyond the area, the calculator also shows:
- Perimeter: the total boundary length,
P = 5a. - Diagonal: the length between two non-adjacent vertices,
d = a × φ. - Circumradius (R): radius of the circumscribed circle.
- Inradius (r): radius of the inscribed circle (apothem).
- Interior angle: always 108° in a regular pentagon.
- Sum of interior angles: always 540° for any pentagon (from the formula (n − 2) × 180°).
Unit Conversion
All length units (mm, cm, m, km, in, ft, yd) are supported for input. The output can be expressed in any area unit (mm², cm², m², km², in², ft², yd²). The calculator converts input lengths to meters internally, computes the area in m², and then converts to the chosen output unit. A conversion table showing the area in every supported unit is displayed alongside the primary result.
Practical Applications
Pentagon area calculations arise in many real-world contexts:
- Architecture and design: Pentagonal floor plans, decorative tiles, and structural cross-sections frequently appear in modern and classical buildings.
- Land surveying: Parcels and property boundaries are sometimes five-sided. The irregular mode with coordinate input directly supports GPS or blueprint data.
- Engineering: Cross-sectional area of pentagonal profiles is needed for stress and load calculations.
- Education: Students can use the step-by-step output to understand polygon area formulas, the golden ratio, and the relationship between a polygon's circumradius, inradius, and side length.
- Game development and graphics: Collision detection and 2D layout algorithms often require fast polygon area computations.
Tips for Accurate Results
For regular pentagon calculations, any one of side length, circumradius, or inradius is sufficient — choose whichever you can measure directly. For irregular pentagons, accurate (x, y) coordinates are essential; even small errors in vertex positions lead to proportional errors in the computed area. Use the precision selector to control how many decimal places are shown in the result.